Archives of: Michigan Logic Seminar

Harry Altman: Lower sets in products of well-ordered sets and related WPOs

Thursday, May 24, 2018, from 4 to 5:30pm
East Hall, room 4096

Speaker: Harry Altman (University of Michigan)

Title: Lower sets in products of well-ordered sets and related WPOs

Abstract:

Following last week’s talk on maximum order types of well partial orders, we’ll compute the maximum order type of the set of bounded lower sets in N^m, as well as generalizations to finite products of other well-ordered sets, and discuss the maximum order types of some other related well partial orders also.

Harry Altman: Well partial orderings and their maximum extending ordinals

Thursday, May 17, 2018, from 4 to 5:30pm
East Hall, room 4096

Speaker: Harry Altman (University of Michigan)

Title: Well partial orderings and their maximum extending ordinals

Abstract:

A well partial order is a partial order all of whose extensions to a total order are well-orders. (These are often studied as well-quasi-orders, where the requirement of antisymmetry is dropped.) In 1976 De Jongh and Parikh showed that for a given WPO X, among the ordinals obtained this way there is always a maximum o(X). We will discuss the theory of WPOs and o(X), several equivalent formulations, and how o(X) can actually be computed for some concrete WPOs.

Andreas Blass: Well-ordered choice implies dependent choice

Thursday, May 3, 2018, from 4 to 5:30pm
East Hall, room 4096

Speaker: Andreas Blass (University of Michigan)

Title: Well-ordered choice implies dependent choice

Abstract:

The axiom of well-ordered choice is a weak form of the axiom of choice. It says that every well-ordered family of nonempty sets has a choice function. The axiom of dependent choice is another weak form of the axiom of choice. It says that, given any directed graph in which every vertex has at least one outgoing arrow, and given any vertex v in that graph, there exists an infinite sequence of vertices that starts at v and then follows the arrows. I’ll prove the old but probably insufficiently well-known theorem of Jensen that well-ordered choice implies dependent choice.

Alexei Kolesnikov: Homology groups in model theory.

Thursday, April 19, 2018, from 4 to 5:30pm
East Hall, room 3088

Speaker: Alexei Kolesnikov (Towson University)

Title: Homology groups in model theory.

Abstract:

Higher-dimensional amalgamation properties played a key role in settling several questions in classification theory. It turns out that these properties, suitably formulated, are non-trivial even for totally categorical first order theories. The main goal of this project was to understand and characterize the failure of higher-dimensional amalgamation properties in stable theories. We show that the failure of n-dimensional amalgamation is detected by a suitable homology group; this group must be abelian profinite and is isomorphic to a certain automorphism group. Along the way, we establish that the failure of n dimensional amalgamation is witnessed by certain canonical objects, with the higher category-theoretic flavor, that are definable in the models of the theory.

Joint work with John Goodrick and Byunghan Kim.

Osvaldo Guzman Gonzalez: On weakly universal functions

Thursday, April 12, 2018, from 4 to 5:30pm
East Hall, room 3088

Speaker: Osvaldo Guzman Gonzalez (York University)

Title: On weakly universal functions

Abstract:

A function U:[omega_1]^2 —> 2 is called universal if for every function F:[omega_1]^2 —> omega there is an injective function h:omega_1 —> omega_1 such that F(alpha,beta)=U(h(alpha),h(beta)) for each alpha,betain omega_1. It is easy to see that universal functions exist assuming the Continuum Hypothesis, furthermore, by results of Shelah and Mekler, the existence of such functions is consistent with the continuum being arbitrarily large. Universal functions were recently studied by Shelah and Steprans, where they showed that the existence of universal graphs is consistent with several values of the dominating and unbounded numbers. They also considered several variations of universal functions, in particular, the following notion was studied: A function U:[omega_1]^2 —> omega is (1,omega_1)-weakly universal if for every F:[omega_1]^2 —> omega there is an injective function h:omega_1 —> omega_1 and a function e:omega —> omega such that F(alpha,beta)=eU(h(alpha),h(beta)) for every alpha,betain omega_1. Shelah and Steprans asked if (1,omega_1)-weakly universal functions exist in ZFC. We will study the existence of (1,omega_1)-weakly universal functions in Sacks models and provide an answer to their problem.

Andreas Blass: Well-Ordered Choice

Thursday, April 5, 2018, from 4 to 5:30pm
East Hall, room 3088

Speaker: Andreas Blass (University of Michigan)

Title: Well-Ordered Choice

Abstract:

The axiom of well-ordered choice is a weak form of the axiom of choice. It says that every well-ordered family of nonempty sets has a choice function. I plan to prove two long-known but perhaps not well-known results about this axiom. The first is the construction of a permutation model (of set theory with atoms) in which the axiom of well-ordered choice holds but the full axiom of choice fails. The second is that well-ordered choice implies the axiom of dependent choice. Dependent choice is the assertion that, given any directed graph in which every vertex has at least one outgoing arrow, and given any vertex v in that graph, there exists an infinite sequence of vertices that starts at v and then follows the arrows. If time permits, I’ll also indicate why that second result is nontrivial, even though dependent choice seems to require only a countable (hence well-ordered) sequence of choices.

David J. Fernández Bretón: Models of set theory with union ultrafilters and small covering of meagre, II

Thursday, March 15, 2018, from 4 to 5:30pm
East Hall, room 3088

Speaker: David J. Fernández Bretón (University of Michigan)

Title: Models of set theory with union ultrafilters and small covering of meagre, II

Abstract:

Union ultrafilters are ultrafilters that arise naturally from Hindman’s finite unions theorem, in much the same way that selective ultrafilters arise from Ramsey’s theorem, and they are very important objects from the perspective of algebra in the Cech–Stone compactification. The existence of union ultrafilters is known to be independent from the ZFC axioms (due to Hindman and Blass–Hindman), and is known to follow from a number of set-theoretic hypothesis, of which the weakest one is that the covering of meagre equals the continuum (this is due to Eisworth). In the first part of this two-talk series I exhibited a model of ZFC with union ultrafilters whose covering of meagre is strictly less than the continuum, obtained by means of a short countable support iteration. In this second talk, I will exhibit two more such models, one obtained by means of a countable support iteration of proper forcings, and the other by means of a single-step forcing (modulo being able to obtain an appropriate ground model).

Dana Bartosova: Ellis’ problem for automorphism groups

Thursday, March 8, 2018, from 4 to 5:30pm
East Hall, room 3088

Speaker: Dana Bartosova (Carnegie Mellon University)

Title: Ellis’ problem for automorphism groups

Abstract:

Ellis’ problem is a problem from topological dynamics asking whether two well studied flows are canonically isomorphic. In the case of groups of automorphisms of discrete structures, we can translate this problem into a question about Boolean algebras and solve the problem for some countable structures. We also arrive at questions about existence of certain ultrafilters. This is a join work with Andy Zucker.

Chris Kapulkin: Homotopy Type Theory and internal languages of higher categories

Thursday, February 22, 2018, from 4 to 5:30pm
East Hall, room 3088

Speaker: Chris Kapulkin (University of Western Ontario)

Title: Homotopy Type Theory and internal languages of higher categories

Abstract:

Homotopy Type Theory (or HoTT) is an approach to foundations of mathematics, building on the homotopy-theoretic interpretation of type theory. In addition to its foundational role, HoTT has been speculated to be the internal language of higher toposes in the sense of Joyal and Lurie.
This talk will be an introduction to HoTT, explaining its main ideas and presenting one way in which the connection between type theory and higher categories can be made precise.

David J. Fernández Bretón: Models of set theory with union ultrafilters and small covering of meagre

Thursday, February 15, 2018, from 4 to 5:30pm
East Hall, room 3088

Speaker: David J. Fernández Bretón (University of Michigan)

Title: Models of set theory with union ultrafilters and small covering of meagre

Abstract:

Union ultrafilters are ultrafilters that arise naturally from Hindman’s finite unions theorem, in much the same way that selective ultrafilters arise from Ramsey’s theorem, and they are very important objects from the perspective of algebra in the Cech–Stone compactification. The existence of union ultrafilters is known to be independent from the ZFC axioms (due to Hindman and Blass–Hindman), and is known to follow from a number of set-theoretic hypothesis, of which the weakest one is that the covering of meagre equals the continuum (this is due to Eisworth). I will show that such hypothesis is not a necessary condition, by exhibiting a number of different models of ZFC that have a covering of meagre strictly less than the continuum, while at the same time satisfying the existence of union ultrafilters.